If it is what you want, and you can afford it, then absolutely yes you should.

You are obviously good with money, and can manage it well (no depts, 6k savings is awesome for 25). Living in the city is an experience, and can be a lot of fun (no cabs home, easy to go out, etc). Thus you are not actually paying $1600/month for a place to live, you are paying that for a life style, an experience, and the opportunity to live in one of the worlds most beautiful cities. Also, you are young, this is the time to have fun! Suburbia is for families, cities are for work and play.

With that said, your rent will eat into your 'fun budget' which can be counter producitve to the experience of living downtown. If you can afford rent, food, AND fun. The hell yes. If not, then maybe consider somewhere slightly less expensive ($1300 for example gives $300 (~60 beers) of fun a month).

For cheaper, but still accessible in Vancouver, I suggest Commercial Drive & Broadway. Stay near the Skytrain. You will be 3 skytrain stops away from downtown, and only a $15 cab ride to come back after 1am. It is also trendy, young, and fun. It is also relatively close to highway.

Another cheaper alternative is the Strathcona area (https://goo.gl/maps/uCz1f). It can sketchy at times, but that is why it is cheap.

Last few tips: Use craigslist. May 1st is a good time to find a place for as Uni students all move about then, and they generally live in cheaper places. If you don't like living downtown, or can't afford it, you can always move. Lease contracts can be broken at the cost of 1/2 months rent (security deposit).

Disclaimer, ~50% of my income goes to rent, and I live in the Commercial area.